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Spectral Gap and Asymptotics for a Family of Cocycles of Perron-Frobenius Operators

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Spectral Gap and Asymptotics for a Family of Cocycles of Perron-Frobenius Operators Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators by Joseph Anthony Horan MSc, University of Victoria, 2015 BMath, University of Waterloo, 2013 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Mathematics and Statistics c Joseph Anthony Horan, 2020 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. We acknowledge with respect the Lekwungen peoples on whose traditional territory the university stands, and the Songhees, Esquimalt, and WSANE´ C´ peoples whose ¯ historical relationships with the land continue to this day. Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators by Joseph Anthony Horan MSc, University of Victoria, 2015 BMath, University of Waterloo, 2013 Supervisory Committee Dr. Christopher Bose, Co-Supervisor (Department of Mathematics and Statistics) Dr. Anthony Quas, Co-Supervisor (Department of Mathematics and Statistics) Dr. Sue Whitesides, Outside Member (Department of Computer Science) ii ABSTRACT At its core, a dynamical system is a set of things and rules for how they change. In the study of dynamical systems, we often ask questions about long-term or average phe- nomena: whether or not there is an equilibrium for the system, and if so, how quickly the system approaches that equilibrium. These questions are more challenging in the non-autonomous (or random) setting, where the rules change over time. The main goal of this dissertation is to develop new tools with which to study random dynamical systems, and demonstrate their application in a non-trivial context. We prove a new Perron-Frobenius theorem for cocycles of bounded linear operators which preserve and sometimes contract a cone in a Banach space; this new theorem provides an explicit up- per bound for the second-largest Lyapunov exponent of the cocycle, which determines how quickly the system approaches its equilibrium-like state. Using this theorem and other tools (including a new Lasota-Yorke-type inequality for Perron-Frobenius opera- tors for use with a family of maps), we show that a class of cocycles of piecewise linear maps has a Lyapunov spectral gap (hence answering the equilibrium question in the affirmative), and we moreover have an explicit lower bound on the spectral gap. We also prove asymptotics for a family of cocycles arising from a perturbation of a fixed map with two invariant densities; we obtain a linear upper bound for the second-largest Lyapunov exponent, and the bound is sharp, in the sense that there are members of this family of perturbations where the second-largest Lyapunov exponent is linear in the perturbation parameter. The sharpness example is studied through an in-depth determinant-free linear algebra computation for Markov operators. iii Table of Contents Supervisory Committee ii Abstract iii Table of Contents iv List of Figures vi Acknowledgements viii Dedication xiv Chapter 1 Introduction 1 Chapter 2 Cocycle Perron-Frobenius Theorem 12 2.1 Cones . 13 2.2 Measurability and Topological Considerations . 30 2.3 Cocycles, Lyapunov Exponents, and the Grassmannian . 33 2.4 Statement of the Main Theorem . 36 2.5 Proof of Theorem 2.4.3 and Corollary 2.4.5 . 39 2.6 Easy Applications of Theorem 2.4.3 . 52 Chapter 3 Balanced Lasota-Yorke-type Inequality 59 3.1 Bounded Variation and Setting . 60 3.2 Statement and Proof of the Inequality . 77 Chapter 4 Application to Cocycles of Perron-Frobenius Operators 91 4.1 Cyclic Decomposition . 94 4.2 Uniform Lasota-Yorke Inequality . 97 4.3 Covering Properties . 104 4.4 Illustration of the Covering Method . 111 iv (n) 4.5 Contraction of the Cone - Spectral Gap for L! . 115 4.6 Perturbation Asymptotics . 126 Chapter 5 Markov Paired Tent Maps 136 5.1 Markov Maps and Partitions . 136 5.2 Spectral Properties of An . 141 5.3 Spectral Properties of a Factor System . 149 5.4 Mixing Rates and Times . 153 5.5 Simultaneous Spectrum via Algebraic Geometry . 155 5.6 Two-Parameter Markov Paired Tent Maps . 160 Chapter 6 Conclusion 164 Appendix A Assorted Lemmas, Proofs, and Computations 166 A.1 Miscellaneous Ergodic Theory . 166 A.2 Miscellaneous Tools . 170 A.3 Miscellaneous Examples . 175 Bibliography 177 v List of Figures Figure 1.1 A Markov chain with four states and its associated transition matrix. 2 Figure 1.2 The stirring map T that moves chocolate chips around in the banana bread batter [−1; 1]. 3 Figure 1.3 The Perron-Frobenius operator L sends densities f to new den- sities Lf (schematic only). 4 Figure 1.4 Schematic of \leaking" behaviour, where 1(!) and 2(!) are gen- erally small. 9 2 Figure 2.1 The cone Cy,α for y = (1; 2) and α = 2 in R , depicted with the perpendicular plane at cy with c = 1=2. 18 Figure 2.2 Schematics of cocycles along an orbit of x (both non-invertible and invertible). 33 3 2 Figure 3.1 FH (f), for f(x) = 36x + 4x − 3x − 0:1 restricted to [−1; 1] and H = [−0:4; 0:3]. 72 Figure 3.2 The setup in Example 3.2.6. 85 Figure 3.3 Computing L(f) using Lemma 3.2.4. 86 Figure 3.4 A map T : [0; 1] ! [0; 1] with four hanging points, namely: (0:4; −), (0:4; +), (0:6; +), and (1; −). 87 Figure 4.1 The paired tent map, with parameters 1 = 0:3 and 2 = 0:7. 92 Figure 4.2 The second iterate of the coupled tent map, S!, with parameters 1(!) = 0:1, 2(!) = 0:2, 1(σ(!)) = 0:1, and 2(σ(!)) = 0:2. 98 Figure 4.3 The second iterate of the coupled tent map, S!, with parameters 1(!) = 0:7, 2(!) = 0:3, 1(σ(!)) = 0:2, and 2(σ(!)) = 0:6. 99 Figure 4.4 The function f and its image under the Perron-Frobenius op- erator P!. Observe that both functions are supported only on [0; 1]. 105 Figure 4.5 The map T in Remark 4.3.6. 109 vi 1 Figure 4.6 A picture of (Lδ1,δ2 − L1,2 ) [−1;−1=2] for 0 < δ1 < 1. The first jump is of size 1 , and the second jump is of size 1 , so 2(1+δ1) 2(1+1) the variation is the sum of the two jump sizes. 126 Figure 5.1 The paired tent map Tκ,κ, with parameter κ = 0:3. 137 Figure 5.2 Markov partitions for Tκn , with n = 1; 4. 140 Figure 5.3 General form of the (2n + 4)-by-(2n + 4) adjacency matrix An. 141 Figure 5.4 The (2n + 4)-by-(2n + 4) matrix Jn. 142 Figure 5.5 A zoomed-in look at the Markov partition for Tn in [−1; 0]. 143 Figure 5.6 Pictures of the roots of fn and gn for different values of n; roots of fn are marked with crosses, roots of gn are marked with circles, and the origin is marked with an asterisk (where An has a double eigenvalue). The circle of radius 2 is a dashed line, the unit circle is a solid line, and the circles with radius 1 ± n−1 are dotted lines.147 ~ Figure 5.7 The map Tκ, for κ = 0:3. 149 ~ Figure 5.8 Markov partitions for Tn, for n = 1; 4. 151 Figure 5.9 General form of the (n + 3)-by-(n + 3) adjacency matrix Bn. 151 Figure 5.10 The (n+3)-by-(n+2) matrix ι, representing the inclusion E+ ! Cn+3.................................. 151 Figure 5.11 The (n + 2)-by-(n + 2) matrix Cn, representing the action of An on E+................................. 152 Figure 5.12 The steeper function (in blue) is dm(κ) and the flatter function (in red) is cn(κ), for m = 2 and n = 3. The x-coordinate of the intersection is the κ value corresponding to (n; m) and the y-coordinate is the ζ value. 161 vii ACKNOWLEDGEMENTS Those who have spent significant time with me over the course of my doctorate will not be surprised to see that my list of acknowledgements is a metaphorical mile long (for the purpose of using metric units, it's actually about 1.15 metres, give or take). For my Master's, I wrote no names; all of the people I am about to name deserve to be recognized for being wonderful people and making my life better in one way or another during my Ph.D. (Note that folks will only be named once, even if they played multiple roles in my life.) If I forgot anyone, that's on me and not on you. First and foremost, I would like to thank my supervisors, Christopher Bose and Anthony Quas. It has been an interesting six years and eight months! Thank you for being fabulous academic mentors. Your willingness to meet with me often, talk about whatever, and be candid with me has been a great help. It is striking how much more I am like Chris than Anthony, isn't it? But I learned so much from both of you regardless. I hope you two appreciated the time we spent together bashing our heads on the chalkboard in the lounge and bothering everyone on the fifth floor with our Skype conversations; I did some good work thinking on the fly, I think. I admit that I will appreciate not having to look your comments on my work for a while, though. I should also express my appreciation for your financial support, through NSERC (in addition to my own scholarship).
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